Ribbon concordance defines an interesting relation on knots. In his initial work on the topic, Gordon asked whether it is a partial order, and this question was open for over 40 years until Agol answered it affirmatively in 2022. However, we still don’t know many basic facts about this partial order: for example, does any infinite chain of ribbon concordances $\dots \leq K_3 \leq K_2 \leq K_1$ eventually stabilize? Even better, if we fix a knot $K$ in the 3-sphere, are there only finitely many knots that are ribbon concordant to $K$? I’ll talk about joint work with John Baldwin toward these questions, in which we use tools from both Heegaard Floer homology and hyperbolic geometry to say that at the very least, there are only finitely many fibered hyperbolic knots ribbon concordant to $K$.
